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| filename
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values | symbolic_name
stringlengths 1
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| docstring
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|---|---|---|---|---|---|---|
odd_p_stable gT p (G : {group gT}) : odd #|G| -> p.-stable G. Proof. move: gT G. pose p_xp gT (E : {group gT}) x := p.-elt x && (x \in 'C([~: E, [set x]])). suffices IH gT (E : {group gT}) x y (G := <<[set x; y]>>) : [&& odd #|G|, p.-group E & G \subset 'N(E)] -> p_xp gT E x && p_xp gT E y -> p.-group (G / 'C(E)). - move=> gT G oddG P A pP /andP[/mulGsubP[_ sPG] _] /andP[sANG pA] cRA. apply/subsetP=> _ /morphimP[x Nx Ax ->]; have NGx := subsetP sANG x Ax. apply: Baer_Suzuki => [|_ /morphimP[y Ny NGy ->]]; first exact: mem_quotient. rewrite -morphJ // -!morphim_set1 -?[<<_>>]morphimY ?sub1set ?groupJ //. set G1 := _ <*> _; rewrite /pgroup -(card_isog (second_isog _)); last first. by rewrite join_subG !sub1set Nx groupJ. have{Nx NGx Ny NGy} [[Gx Nx] [Gy Ny]] := (setIP NGx, setIP NGy). have sG1G: G1 \subset G by rewrite join_subG !sub1set groupJ ?andbT. have nPG1: G1 \subset 'N(P) by rewrite join_subG !sub1set groupJ ?andbT. rewrite -setIA setICA (setIidPr sG1G). rewrite (card_isog (second_isog _)) ?norms_cent //. apply: IH => //; first by rewrite pP nPG1 (oddSg sG1G). rewrite /p_xp -{2}(normP Ny) -conjg_set1 -conjsRg centJ memJ_conjg. rewrite p_eltJ andbb (mem_p_elt pA) // -sub1set centsC (sameP commG1P trivgP). by rewrite -cRA !commgSS ?sub1set. move: {2}_.+1 (ltnSn #|E|) => n; elim: n => // n IHn in gT E x y G *. rewrite ltnS => leEn /and3P[oddG pE nEG] /and3P[/andP[p_x cRx] p_y cRy]. have [Gx Gy]: x \in G /\ y \in G by apply/andP; rewrite -!sub1set -join_subG. apply: wlog_neg => p'Gc; apply/pgroupP=> q q_pr qGc; apply/idPn => p'q. have [Q sylQ] := Sylow_exists q [group of G]. have [sQG qQ]: Q \subset G /\ q.-group Q by case/and3P: sylQ. have{qQ p'q} p'Q: p^'.-group Q by apply: sub_in_pnat qQ => q' _ /eqnP->. have{q q_pr sylQ qGc} ncEQ: ~~ (Q \subset 'C(E)). apply: contraL qGc => cEQ; rewrite -p'natE // -partn_eq1 //. have nCQ: Q \subset 'N('C(E)) by apply: subset_trans (normG _). have sylQc: q.-Sylow(G / 'C(E)) (Q / 'C(E)) by rewrite morphim_pSylow. by rewrite -(card_Hall sylQc) -trivg_card1 (sameP eqP trivgP) quotient_sub1. have solE: solvable E := pgroup_sol pE. have ntE: E :!=: 1 by apply: contra ncEQ; move/eqP->; rewrite cents1. have{Q ncEQ p'Q sQG} minE_EG: minnormal E (E <*> G). apply/mingroupP; split=> [|D]; rewrite join_subG ?ntE ?normG //. case/and3P=> ntD nDE nDG sDE; have nDGi := subsetP nDG. apply/eqP; rewrite eqEcard sDE leqNgt; apply: contra ncEQ => ltDE. have nDQ: Q \subset 'N(D) by rewrite (subset_trans sQG). have cDQ: Q \subset 'C(D). rewrite -quotient_sub1 ?norms_cent // ?[_ / _]card1_trivg //. apply: pnat_1 (morphim_pgroup _ p'Q); apply: pgroupS (quotientS _ sQG) _. apply: IHn (leq_trans ltDE leEn) _ _; first by rewrite oddG (pgroupS sDE). rewrite /p_xp p_x p_y /=; apply/andP. by split; [move: cRx | move: cRy]; apply: subsetP; rewrite centS ?commSg. apply: (stable_factor_cent cDQ) solE; rewrite ?(pnat_coprime pE) //. apply/and3P; split; rewrite // -quotient_cents2 // centsC. rewrite -quotient_sub1 ?norms_cent ?quotient_norms ?(subset_trans sQG) //=. rewrite [(_ / _) / _]card1_trivg //=. apply: pnat_1 (morphim_pgroup _ (morphim_pgroup _ p'Q)). apply: pgroupS (quotientS _ (quotientS _ sQG)) _. have defGq: G / D = <<[set coset D x; coset D y]>>. by rewrite quotient_gen -1?gen_subG ?quotientU ?quotient_set1 ?nDGi. rewrite /= defGq IHn ?(leq_trans _ leEn) ?ltn_quotient // -?defGq. by rewrite quotient_odd // quotient_pgroup // quotient_norms. rewrite /p_xp -!sub1set !morph_p_elt -?quotient_set1 ?nDGi //=. by rewrite -!quotientR ?quotient_cents ?sub1set ?nDGi. have abelE: p.-abelem E. by rewrite -is_abelem_pgroup //; case: (minnormal_solvable minE_EG _ solE). have cEE: abelian E by case/and3P: abelE. have{minE_EG} minE: minnormal E G. case/mingroupP: minE_EG => _ minE; apply/mingroupP; rewrite ntE. split=> // D ntD sDE; apply: minE => //; rewrite join_subG cents_norm //. by rewrite centsC (subset_trans sDE). have nCG: G \subset 'N('C_G(E)) by rewrite normsI ?normG ?norms_cent. suffices{p'Gc} pG'c: p.-group (G / 'C_G(E))^`(1). have [Pc sylPc sGc'Pc]:= Sylow_superset (der_subS _ _) pG'c. have nsPc: Pc <| G / 'C_G(E) by rewrite sub_der1_normal ?(pHall_sub sylPc). case/negP: p'Gc; rewrite /pgroup -(card_isog (second_isog _)) ?norms_cent //. rewrite setIC; apply: pgroupS (pHall_pgroup sylPc) => /=. rewrite sub_quotient_pre // join_subG !sub1set !(subsetP nCG, inE) //=. by rewrite !(mem_normal_Hall sylPc) ?mem_quotient ?morph_p_elt ?(subsetP nCG). have defC := rker_abelem abelE ntE nEG; rewrite /= -/G in defC. set rG := abelem_repr abelE ntE nEG in defC. case ncxy: (rG x *m rG y == rG y *m rG x). have Cxy: [~ x, y] \in 'C_G(E). rewrite -defC inE groupR //= !repr_mxM ?groupM ?groupV // mul1mx -/rG. by rewrite (eqP ncxy) -!repr_mxM ?groupM ?groupV // mulKg mulVg repr_mx1. rewrite [_^`(1)](commG1P _) ?pgroup1 //= quotient_gen -gen_subG //= -/G. rewrite !gen_subG centsC gen_subG quotient_cents2r ?gen_subG //= -/G. rewrite /commg_set imset2Ul !imset2_set1l !imsetU !imset_set1. by rewrite !subUset andbC !sub1set !commgg group1 /= -invg_comm groupV Cxy. pose Ax : 'M(E) := rG x - 1; pose Ay : 'M(E) := rG y - 1. have Ax2: Ax *m Ax = 0. apply/row_matrixP=> i; apply/eqP; rewrite row_mul mulmxBr mulmx1. rewrite row0 subr_eq0 -(can_eq (rVabelemK abelE ntE)) rVabelemJ //. rewrite conjgE -(centP cRx) ?mulKg //. rewrite linearB /= addrC row1 rowE rVabelemD rVabelemN rVabelemJ //=. by rewrite mem_commg ?set11 ?mem_rVabelem. have Ay2: Ay *m Ay = 0. apply/row_matrixP=> i; apply/eqP; rewrite row_mul mulmxBr mulmx1. rewrite row0 subr_eq0 -(inj_eq (@rVabelem_inj _ _ _ abelE ntE)). rewrite rVabelemJ // conjgE -(centP cRy) ?mulKg //. rewrite linearB /= addrC row1 rowE rVabelemD rVabelemN rVabelemJ //=. by rewrite mem_commg ?set11 ?mem_rVabelem. pose A := Ax *m Ay + Ay *m Ax. have cAG: centgmx rG A. rewrite /centgmx gen_subG subUset !sub1set !inE Gx Gy /=; apply/andP. rewrite -[rG x](subrK 1%R) -[rG y](subrK 1%R) -/Ax -/Ay. rewrite 2!(mulmxDl _ 1 A) 2!(mulmxDr A _ 1) !mulmx1 !mul1mx. rewrite !(inj_eq (addIr A)) ![_ *m A]mulmxDr ![A *m _]mulmxDl. by rewrite -!mulmxA Ax2 Ay2 !mulmx0 !mulmxA Ax2 Ay2 !mul0mx !addr0 !add0r. have irrG: mx_irreducible rG by apply/abelem_mx_irrP. pose FA := gen_of irrG cAG; pose dA := gen_dim A. pose rAG : mx_representation FA G dA := gen_repr irrG cAG. pose inFA m W : 'M[FA]_(m, dA) := in_gen irrG cAG W. pose valFA m (W : 'M[FA]_(m, dA)) := val_gen W. rewrite -(rker_abelem abelE ntE nEG) -/rG -(rker_gen irrG cAG) -/rAG. have dA_gt0: dA > 0 by rewrite (gen_dim_gt0 irrG cAG). have irrAG: mx_irreducible rAG by apply: gen_mx_irr. have: dA <= 2. case Ax0: (Ax == 0). by rewrite subr_eq0 in Ax0; case/eqP: ncxy; rewrite (eqP Ax0) mulmx1 mul1mx. case/rowV0Pn: Ax0 => v /submxP[u def_v nzv]. pose U := col_mx v (v *m Ay); pose UA := <<inFA (1 + 1)%N U>>%MS. pose rvalFA m (W : 'M[FA]_(m, dA)) := rowval_gen W. have Umod: mxmodule rAG UA. rewrite /mxmodule gen_subG subUset !sub1set !inE Gx Gy /= andbC. apply/andP; split; rewrite (eqmxMr _ (genmxE _)) -in_genJ // genmxE. rewrite submx_in_gen // -[rG y](subrK 1%R) -/Ay mulmxDr mulmx1. rewrite addmx_sub // mul_col_mx -mulmxA Ay2 mulmx0. by rewrite -!addsmxE addsmx0 addsmxSr. rewrite -[rG x](subrK 1%R) -/Ax mulmxDr mulmx1 in_genD mul_col_mx. rewrite -mulmxA -[Ay *m Ax](addKr (Ax *m Ay)) (mulmxDr v _ A) -mulmxN. rewrite mulmxA {1 2}def_v -(mulmxA u) Ax2 mulmx0 mul0mx add0r. pose B := A; rewrite -(mul0mx _ B) -mul_col_mx -[B](mxval_groot irrG cAG). rewrite {B} -[_ 0 v](in_genK irrG cAG) -val_genZ val_genK. rewrite addmx_sub ?scalemx_sub ?submx_in_gen //. by rewrite -!addsmxE adds0mx addsmxSl. have nzU: UA != 0. rewrite -mxrank_eq0 genmxE mxrank_eq0 -(can_eq (fun W => val_genK W)). by rewrite in_genK val_gen0 -submx0 col_mx_sub submx0 negb_and nzv. case/mx_irrP: irrAG => _ /(_ UA Umod nzU)/eqnP <-. by rewrite genmxE rank_leq_row. rewrite leq_eqVlt ltnS leq_eqVlt ltnNge dA_gt0 orbF orbC; case/pred2P=> def_dA. rewrite [_^`(1)](commG1P _) ?pgroup1 // quotient_cents2r // gen_subG. apply/subsetP=> zt; case/imset2P=> z t Gz Gt ->{zt}. rewrite !inE groupR //= mul1mx; have Gtz := groupM Gt Gz. rewrite -(inj_eq (can_inj (mulKmx (repr_mx_unit rAG Gtz)))) mulmx1. rewrite [eq_op]lock -repr_mxM ?groupR ?groupM // -commgC !repr_mxM // -lock. apply/eqP; move: (rAG z) (rAG t); rewrite /= -/dA def_dA => Az At. by rewrite [Az]mx11_scalar scalar_mxC. move: (kquo_repr _) (kquo_mx_faithful rAG) => /=; set K := rker _. rewrite def_dA => r2G; move/der1_odd_GL2_charf; move/implyP. rewrite quotient_odd //= -/G; apply: etrans; apply: eq_pgroup => p'. have [p_pr _ _] := pgroup_pdiv pE ntE. by rewrite (fmorph_char (gen _ _)) (charf_eq (char_Fp _)). Qed.
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
odd_p_stable
| |
C := 'C_G(P).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
C
| |
defN : 'N_G(P) = G. Proof. by rewrite (setIidPl _) ?normal_norm. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
defN
| |
nsCG : C <| G. Proof. by rewrite -defN subcent_normal. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
nsCG
| |
nCG := normal_norm nsCG.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
nCG
| |
nCX := subset_trans sXG nCG. (* This is B & G, Theorem A.5.1; it does not depend on the solG assumption. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
nCX
| |
odd_abelian_gen_stable : X / C \subset 'O_p(G / C). Proof. case/exists_eqP: genX => gX defX. rewrite -defN sub_quotient_pre // -defX gen_subG. apply/bigcupsP=> A gX_A; have [_ pA nAP cAA] := and4P gX_A. have{gX_A} sAX: A \subset X by rewrite -defX sub_gen ?bigcup_sup. rewrite -sub_quotient_pre ?(subset_trans sAX nCX) //=. rewrite odd_p_stable ?normalM ?pcore_normal //. by rewrite /psubgroup pA defN (subset_trans sAX sXG). by apply/commG1P; rewrite (subset_trans _ cAA) // commg_subr. Qed. (* This is B & G, Theorem A.5.2. *)
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
odd_abelian_gen_stable
| |
odd_abelian_gen_constrained : 'O_p^'(G) = 1 -> 'C_('O_p(G))(P) \subset P -> X \subset 'O_p(G). Proof. set Q := 'O_p(G) => p'G1 sCQ_P. have sPQ: P \subset Q by rewrite pcore_max. have defQ: 'O_{p^', p}(G) = Q by rewrite pseries_pop2. have pQ: p.-group Q by apply: pcore_pgroup. have sCQ: 'C_G(Q) \subset Q. by rewrite -{2}defQ solvable_p_constrained //= defQ /pHall pQ indexgg subxx. have pC: p.-group C. apply/pgroupP=> q q_pr; case/Cauchy=> // u Cu q_u; apply/idPn=> p'q. suff cQu: u \in 'C_G(Q). case/negP: p'q; have{q_u}: q %| #[u] by rewrite q_u. by apply: pnatP q q_pr => //; apply: mem_p_elt pQ _; apply: (subsetP sCQ). have [Gu cPu] := setIP Cu; rewrite inE Gu /= -cycle_subG. rewrite coprime_nil_faithful_cent_stab ?(pgroup_nil pQ) //= -/C -/Q. - by rewrite cycle_subG; apply: subsetP Gu; rewrite normal_norm ?pcore_normal. - by rewrite (pnat_coprime pQ) // [#|_|]q_u pnatE. have sPcQu: P \subset 'C_Q(<[u]>) by rewrite subsetI sPQ centsC cycle_subG. by apply: subset_trans (subset_trans sCQ_P sPcQu); rewrite setIS // centS. rewrite -(quotientSGK nCX) ?pcore_max // -pquotient_pcore //. exact: odd_abelian_gen_stable. Qed.
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
odd_abelian_gen_constrained
| |
Puig_char G : 'L(G) \char G. Proof. exact: gFchar. Qed.
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_char
| |
center_Puig_char G : 'Z('L(G)) \char G. Proof. by rewrite !gFchar_trans. Qed. (* This is B & G, Lemma B.1(a). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
center_Puig_char
| |
Puig_succS G D E : D \subset E -> 'L_[G](E) \subset 'L_[G](D). Proof. move=> sDE; apply: Puig_max (Puig_succ_sub _ _). exact: norm_abgenS sDE (Puig_gen _ _). Qed. (* This is part of B & G, Lemma B.1(b) (see also BGsection1.Puig1). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_succS
| |
Puig_sub_even m n G : m <= n -> 'L_{m.*2}(G) \subset 'L_{n.*2}(G). Proof. move/subnKC <-; move: {n}(n - m)%N => n. by elim: m => [|m IHm] /=; rewrite ?sub1G ?Puig_succS. Qed. (* This is part of B & G, Lemma B.1(b). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_sub_even
| |
Puig_sub_odd m n G : m <= n -> 'L_{n.*2.+1}(G) \subset 'L_{m.*2.+1}(G). Proof. by move=> le_mn; rewrite Puig_succS ?Puig_sub_even. Qed. (* This is part of B & G, Lemma B.1(b). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_sub_odd
| |
Puig_sub_even_odd m n G : 'L_{m.*2}(G) \subset 'L_{n.*2.+1}(G). Proof. elim: n m => [|n IHn] m; first by rewrite Puig1 Puig_at_sub. by case: m => [|m]; rewrite ?sub1G ?Puig_succS ?IHn. Qed. (* This is B & G, Lemma B.1(c). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_sub_even_odd
| |
Puig_limit G : exists m, forall k, m <= k -> 'L_{k.*2}(G) = 'L_*(G) /\ 'L_{k.*2.+1}(G) = 'L(G). Proof. pose L2G m := 'L_{m.*2}(G); pose n := #|G|. have []: #|L2G n| <= n /\ n <= n by rewrite subset_leq_card ?Puig_at_sub. elim: {1 2 3}n => [| m IHm leLm1 /ltnW]; first by rewrite leqNgt cardG_gt0. have [eqLm le_mn|] := eqVneq (L2G m.+1) (L2G m); last first. rewrite eq_sym eqEcard Puig_sub_even ?leqnSn // -ltnNge => lt_m1_m. exact: IHm (leq_trans lt_m1_m leLm1). have{} eqLm k: m <= k -> 'L_{k.*2}(G) = L2G m. rewrite leq_eqVlt => /predU1P[-> // |]; elim: k => // k IHk. by rewrite leq_eqVlt => /predU1P[<- //| ltmk]; rewrite -eqLm !PuigS IHk. by exists m => k le_mk; rewrite Puig_def PuigS /Puig_inf /= !eqLm. Qed. (* This is B & G, Lemma B.1(d), second part; the first part is covered by *) (* BGsection1.Puig_inf_sub. *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_limit
| |
Puig_inf_sub_Puig G : 'L_*(G) \subset 'L(G). Proof. exact: Puig_sub_even_odd. Qed. (* This is B & G, Lemma B.1(e). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_inf_sub_Puig
| |
abelian_norm_Puig n G A : n > 0 -> abelian A -> A <| G -> A \subset 'L_{n}(G). Proof. case: n => // n _ cAA /andP[sAG nAG]. rewrite PuigS sub_gen // bigcup_sup // inE sAG /norm_abelian cAA andbT. exact: subset_trans (Puig_at_sub n G) nAG. Qed. (* This is B & G, Lemma B.1(f), first inclusion. *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
abelian_norm_Puig
| |
sub_cent_Puig_at n p G : n > 0 -> p.-group G -> 'C_G('L_{n}(G)) \subset 'L_{n}(G). Proof. move=> n_gt0 pG. have /ex_maxgroup[M /(max_SCN pG)SCN_M]: exists M, (gval M <| G) && abelian M. by exists 1%G; rewrite normal1 abelian1. have{SCN_M} [cMM [nsMG defCM]] := (SCN_abelian SCN_M, SCN_P SCN_M). have sML: M \subset 'L_{n}(G) by apply: abelian_norm_Puig. by apply: subset_trans (sML); rewrite -defCM setIS // centS. Qed. (* This is B & G, Lemma B.1(f), second inclusion. *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sub_cent_Puig_at
| |
sub_center_cent_Puig_at n G : 'Z(G) \subset 'C_G('L_{n}(G)). Proof. by rewrite setIS ?centS ?Puig_at_sub. Qed. (* This is B & G, Lemma B.1(f), third inclusion (the fourth is trivial). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sub_center_cent_Puig_at
| |
sub_cent_Puig_inf p G : p.-group G -> 'C_G('L_*(G)) \subset 'L_*(G). Proof. by apply: sub_cent_Puig_at; rewrite double_gt0. Qed. (* This is B & G, Lemma B.1(f), fifth inclusion (the sixth is trivial). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sub_cent_Puig_inf
| |
sub_cent_Puig p G : p.-group G -> 'C_G('L(G)) \subset 'L(G). Proof. exact: sub_cent_Puig_at. Qed. (* This is B & G, Lemma B.1(f), final remark (we prove the contrapositive). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sub_cent_Puig
| |
trivg_center_Puig_pgroup p G : p.-group G -> 'Z('L(G)) = 1 -> G :=: 1. Proof. move=> pG LG1; apply/(trivg_center_pgroup pG)/trivgP. rewrite -(trivg_center_pgroup (pgroupS (Puig_sub _) pG) LG1). by apply: subset_trans (sub_cent_Puig pG); apply: sub_center_cent_Puig_at. Qed. (* This is B & G, Lemma B.1(g), second part; the first part is simply the *) (* definition of 'L(G) in terms of 'L_*(G). *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
trivg_center_Puig_pgroup
| |
Puig_inf_def G : 'L_*(G) = 'L_[G]('L(G)). Proof. have [k defL] := Puig_limit G. by case: (defL k) => // _ <-; case: (defL k.+1) => [|<- //]; apply: leqnSn. Qed. (* This is B & G, Lemma B.2. *)
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_inf_def
| |
sub_Puig_eq G H : H \subset G -> 'L(G) \subset H -> 'L(H) = 'L(G). Proof. move=> sHG sLG_H; apply/setP/subset_eqP/andP. have sLH_G := subset_trans (Puig_succ_sub _ _) sHG. have gPuig := norm_abgenS _ (Puig_gen _ _). have [[kG defLG] [kH defLH]] := (Puig_limit G, Puig_limit H). have [/defLG[_ {1}<-] /defLH[_ <-]] := (leq_maxl kG kH, leq_maxr kG kH). split; do [elim: (maxn _ _) => [|k IHk /=]; first by rewrite !Puig1]. rewrite doubleS !(PuigS _.+1) Puig_max ?gPuig // Puig_max ?gPuig //. exact: subset_trans (Puig_sub_even_odd _.+1 _ _) sLG_H. rewrite doubleS Puig_max // -!PuigS Puig_def gPuig //. by rewrite Puig_inf_def Puig_max ?gPuig ?sLH_G. Qed.
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sub_Puig_eq
| |
norm_abgen_pgroup p X G : p.-group G -> X --> G -> generated_by (p_norm_abelian p X) G. Proof. move=> pG /exists_eqP[gG defG]. have:= subxx G; rewrite -{1 3}defG gen_subG /= => /bigcupsP-sGG. apply/exists_eqP; exists gG; congr <<_>>; apply: eq_bigl => A. by rewrite andbA andbAC andb_idr // => /sGG/pgroupS->. Qed. Variables (p : nat) (G S : {group gT}). Hypotheses (oddG : odd #|G|) (solG : solvable G) (sylS : p.-Sylow(G) S). Hypothesis p'G1 : 'O_p^'(G) = 1.
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
norm_abgen_pgroup
| |
T := 'O_p(G).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
T
| |
nsTG : T <| G := pcore_normal _ _.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
nsTG
| |
pT : p.-group T := pcore_pgroup _ _.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
pT
| |
pS : p.-group S := pHall_pgroup sylS.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
pS
| |
sSG := pHall_sub sylS. (* This is B & G, Lemma B.3. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
sSG
| |
pcore_Sylow_Puig_sub : 'L_*(S) \subset 'L_*(T) /\ 'L(T) \subset 'L(S). Proof. have [[kS defLS] [kT defLT]] := (Puig_limit S, Puig_limit [group of T]). have [/defLS[<- <-] /defLT[<- <-]] := (leq_maxl kS kT, leq_maxr kS kT). have sL_ := subset_trans (Puig_succ_sub _ _). elim: (maxn kS kT) => [|k [_ sL1]]; first by rewrite !Puig1 pcore_sub_Hall. have{sL1} gL: 'L_{k.*2.+1}(T) --> 'L_{k.*2.+2}(S). exact: norm_abgenS sL1 (Puig_gen _ _). have sCT_L: 'C_T('L_{k.*2.+1}(T)) \subset 'L_{k.*2.+1}(T). exact: sub_cent_Puig_at pT. have{sCT_L} sLT: 'L_{k.*2.+2}(S) \subset T. apply: odd_abelian_gen_constrained sCT_L => //. - exact: pgroupS (Puig_at_sub _ _) pT. - exact: gFnormal_trans nsTG. - exact: sL_ sSG. by rewrite norm_abgen_pgroup // (pgroupS _ pS) ?Puig_at_sub. have sL2: 'L_{k.*2.+2}(S) \subset 'L_{k.*2.+2}(T) by apply: Puig_max. split; [exact: sL2 | rewrite doubleS; apply: subset_trans (Puig_succS _ sL2) _]. by rewrite Puig_max -?PuigS ?Puig_gen // sL_ // pcore_sub_Hall. Qed.
|
Lemma
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
pcore_Sylow_Puig_sub
| |
Y := 'Z('L(T)).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Y
| |
L := 'L(S). (* This is B & G, Theorem B.4(b). *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
L
| |
Puig_center_normal : 'Z(L) <| G. Proof. have [sLiST sLTS] := pcore_Sylow_Puig_sub. have sLiLT: 'L_*(T) \subset 'L(T) by apply: Puig_sub_even_odd. have sZY: 'Z(L) \subset Y. rewrite subsetI andbC subIset ?centS ?orbT //=. suffices: 'C_S('L_*(S)) \subset 'L(T). by apply: subset_trans; rewrite setISS ?Puig_sub ?centS ?Puig_sub_even_odd. apply: subset_trans (subset_trans sLiST sLiLT). by apply: sub_cent_Puig_at pS; rewrite double_gt0. have chY: Y \char G by rewrite !gFchar_trans. have nsCY_G: 'C_G(Y) <| G by rewrite char_normal 1?subcent_char ?char_refl. have [C defC sCY_C nsCG] := inv_quotientN nsCY_G (pcore_normal p _). have sLG: L \subset G by rewrite gFsub_trans ?(pHall_sub sylS). have nsL_nCS: L <| 'N_G(C :&: S). have sYLiS: Y \subset 'L_*(S). rewrite abelian_norm_Puig ?double_gt0 ?center_abelian //. apply: normalS (pHall_sub sylS) (char_normal chY). by rewrite subIset // (subset_trans sLTS) ?Puig_sub. have gYL: Y --> L := norm_abgenS sYLiS (Puig_gen _ _). have sLCS: L \subset C :&: S. rewrite subsetI Puig_sub andbT. rewrite -(quotientSGK _ sCY_C) ?(subset_trans sLG) ?normal_norm // -defC. rewrite odd_abelian_gen_stable ?char_normal ?norm_abgen_pgroup //. by rewrite (pgroupS _ pT) ?subIset // Puig_sub. by rewrite (pgroupS _ pS) ?Puig_sub. rewrite -[L](sub_Puig_eq _ sLCS) ?subsetIr // gFnormal_trans ?normalSG //. by rewrite subIset // sSG orbT. have sylCS: p.-Sylow(C) (C :&: S) := Sylow_setI_normal nsCG sylS. have{} defC: 'C_G(Y) * (C :&: S) = C. apply/eqP; rewrite eqEsubset mulG_subG sCY_C subsetIl /=. have nCY_C: C \subset 'N('C_G(Y)). exact: subset_trans (normal_sub nsCG) (normal_norm nsCY_G). rewrite -quotientSK // -defC /= -pseries1. rewrite -(pseries_catr_id [:: p : nat_pred]) (pseries_rcons_id [::]) /=. rewrite pseries1 /= pseries1 defC pcore_sub_Hall // morphim_pHall //. by rewrite subIset ?nCY_C. have defG: 'C_G(Y) * 'N_G(C :&: S) = G. have sCS_N: C :&: S \subset 'N_G(C :&: S). by rewrite subsetI normG subIset // sSG orbT. by rewrite -(mulSGid sCS_N) mulgA defC (Frattini_arg _ sylCS). have nsZ_N: 'Z(L) <| 'N_G(C :&: S) := gFnormal_trans _ nsL_nCS. rewrite /normal subIset ?sLG //= -{1}defG mulG_subG /=. rewrite cents_norm ?normal_norm // centsC. by rewrite (subset_trans sZY) // centsC subsetIr. Qed.
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_center_normal
| |
Puig_factorization : 'O_p^'(G) * 'N_G('Z('L(S))) = G. Proof. set D := 'O_p^'(G); set Z := 'Z(_); have [sSG pS _] := and3P sylS. have sSN: S \subset 'N(D) by rewrite (subset_trans sSG) ?gFnorm. have p'D: p^'.-group D := pcore_pgroup _ _. have tiSD: S :&: D = 1 := coprime_TIg (pnat_coprime pS p'D). have def_Zq: Z / D = 'Z('L(S / D)). rewrite !quotientE -(setIid S) -(morphim_restrm sSN); set f := restrm _ _. have injf: 'injm f by rewrite ker_restrm ker_coset tiSD. rewrite -!(injmF _ injf) ?Puig_sub //= morphim_restrm. by rewrite (setIidPr _) // subIset ?Puig_sub. have{def_Zq} nZq: Z / D <| G / D. have sylSq: p.-Sylow(G / D) (S / D) by apply: morphim_pHall. rewrite def_Zq (Puig_center_normal _ _ sylSq) ?quotient_odd ?quotient_sol //. exact: trivg_pcore_quotient. have sZS: Z \subset S by rewrite subIset ?Puig_sub. have sZN: Z \subset 'N_G(Z) by rewrite subsetI normG (subset_trans sZS). have nDZ: Z \subset 'N(D) by rewrite (subset_trans sZS). rewrite -(mulSGid sZN) mulgA -(norm_joinEr nDZ) (@Frattini_arg p) //= -/D -/Z. rewrite -cosetpre_normal quotientK ?quotientGK ?pcore_normal // in nZq. by rewrite norm_joinEr. rewrite /pHall -divgS joing_subr ?(pgroupS sZS) /= ?norm_joinEr //= -/Z. by rewrite TI_cardMg ?mulnK //; apply/trivgP; rewrite /= setIC -tiSD setSI. Qed.
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism automorphism quotient.",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra.",
"From mathcomp Require Import center gfunctor commutator gseries pgroup.",
"From mathcomp Require Import nilpotent sylow abelian maximal.",
"From mathcomp Require Import mxrepresentation mxabelem.",
"From odd_order Require Import BGsection1 BGsection2."
] |
theories/BGappendixAB.v
|
Puig_factorization
| |
nU := ((p ^ q).-1 %/ p.-1)%N. (* External statement of the finite field assumption. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
nU
| |
finFieldImage : Prop := FinFieldImage (F : finFieldType) (sigma : {morphism P >-> F}) of isom P [set: F] sigma & sigma @*^-1 <[1%R : F]> = P0 & exists2 sigmaU : {morphism U >-> {unit F}}, 'injm sigmaU & {in P & U, morph_act 'J 'U sigma sigmaU}. (* These correspond to hypothesis (A) of B & G, Appendix C, Theorem C. *) Hypotheses (pr_p : prime p) (pr_q : prime q) (coUp1 : coprime nU p.-1). Hypotheses (defH : P ><| U = H) (fieldH : finFieldImage). Hypotheses (oP : #|P| = (p ^ q)%N) (oU : #|U| = nU). (* These correspond to hypothesis (B) of B & G, Appendix C, Theorem C. *) Hypotheses (abelQ : q.-abelem Q) (nQP0 : P0 \subset 'N(Q)). Hypothesis nU_P0Q : exists2 y, y \in Q & P0 :^ y \subset 'N(U).
|
Variant
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
finFieldImage
| |
Fpq : {vspace F} := fullv.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
Fpq
| |
Fp : {vspace F} := 1%VS. Hypothesis oF : #|F| = (p ^ q)%N.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
Fp
| |
oF_p : #|'F_p| = p. Proof. exact: card_Fp. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
oF_p
| |
oFp : #|Fp| = p. Proof. by rewrite (@card_vspace1 _ _ (Falgebra.class (PrimeCharType _))). Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
oFp
| |
oFpq : #|Fpq| = (p ^ q)%N. Proof. by rewrite card_vspacef. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
oFpq
| |
dimFpq : \dim Fpq = q. Proof. by rewrite primeChar_dimf oF pfactorK. Qed. Variables (sigma : {morphism P >-> F}) (sigmaU : {morphism U >-> {unit F}}). Hypotheses (inj_sigma : 'injm sigma) (inj_sigmaU : 'injm sigmaU). Hypothesis im_sigma : sigma @* P = [set: F]. Variable s : gT. Hypotheses (sP0P : P0 \subset P) (sigma_s : sigma s = 1) (defP0 : <[s]> = P0).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
dimFpq
| |
psi u : F := val (sigmaU u).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psi
| |
inj_psi : {in U &, injective psi}. Proof. by move=> u v Uu Uv /val_inj/(injmP inj_sigmaU)->. Qed. Hypothesis sigmaJ : {in P & U, forall x u, sigma (x ^ u) = sigma x * psi u}.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
inj_psi
| |
Ps : s \in P. Proof. by rewrite -cycle_subG defP0. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
Ps
| |
P0s : s \in P0. Proof. by rewrite -defP0 cycle_id. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
P0s
| |
nz_psi u : psi u != 0. Proof. by rewrite -unitfE (valP (sigmaU u)). Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
nz_psi
| |
sigma1 : sigma 1%g = 0. Proof. exact: morph1. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigma1
| |
sigmaM : {in P &, {morph sigma : x1 x2 / (x1 * x2)%g >-> x1 + x2}}. Proof. exact: morphM. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigmaM
| |
sigmaV : {in P, {morph sigma : x / x^-1%g >-> - x}}. Proof. exact: morphV. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigmaV
| |
sigmaX n : {in P, {morph sigma : x / (x ^+ n)%g >-> x *+ n}}. Proof. exact: morphX. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigmaX
| |
psi1 : psi 1%g = 1. Proof. by rewrite /psi morph1. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psi1
| |
psiM : {in U &, {morph psi : u1 u2 / (u1 * u2)%g >-> u1 * u2}}. Proof. by move=> u1 u2 Uu1 Uu2; rewrite /psi morphM. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psiM
| |
psiV : {in U, {morph psi : u / u^-1%g >-> u^-1}}. Proof. by move=> u Uu; rewrite /psi morphV. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psiV
| |
psiX n : {in U, {morph psi : u / (u ^+ n)%g >-> u ^+ n}}. Proof. by move=> u Uu; rewrite /psi morphX // val_unitX. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psiX
| |
sigmaE := (sigma1, sigma_s, mulr1, mul1r, (sigmaJ, sigmaX, sigmaM, sigmaV), (psi1, psiX, psiM, psiV)).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigmaE
| |
psiE u : u \in U -> psi u = sigma (s ^ u). Proof. by move=> Uu; rewrite !sigmaE. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
psiE
| |
nPU : U \subset 'N(P). Proof. by have [] := sdprodP defH. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
nPU
| |
memJ_P : {in P & U, forall x u, x ^ u \in P}. Proof. by move=> x u Px Uu; rewrite /= memJ_norm ?(subsetP nPU). Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
memJ_P
| |
in_PU := (memJ_P, in_group).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
in_PU
| |
sigmaP0 : sigma @* P0 =i Fp. Proof. rewrite -defP0 morphim_cycle // sigma_s => x. apply/cycleP/vlineP=> [] [n ->]; first by exists n%:R; rewrite scaler_nat. by exists (val n); rewrite -{1}[n]natr_Zp -in_algE rmorph_nat zmodXgE. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sigmaP0
| |
nt_s : s != 1%g. Proof. by rewrite -(morph_injm_eq1 inj_sigma) // sigmaE oner_eq0. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
nt_s
| |
p_gt0 : (0 < p)%N. Proof. exact: prime_gt0. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
p_gt0
| |
q_gt0 : (0 < q)%N. Proof. exact: prime_gt0. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
q_gt0
| |
p1_gt0 : (0 < p.-1)%N. Proof. by rewrite -subn1 subn_gt0 prime_gt1. Qed. (* This is B & G, Appendix C, Remark I. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
p1_gt0
| |
not_dvd_q_p1 : ~~ (q %| p.-1)%N. Proof. rewrite -prime_coprime // -[q]card_ord -sum1_card -coprime_modl -modn_summ. have:= coUp1; rewrite /nU predn_exp mulKn //= -coprime_modl -modn_summ. congr (coprime (_ %% _) _); apply: eq_bigr => i _. by rewrite -{1}[p](subnK p_gt0) subn1 -modnXm modnDl modnXm exp1n. Qed. (* This is the first assertion of B & G, Appendix C, Remark V. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
not_dvd_q_p1
| |
odd_p : odd p. Proof. by apply: contraLR ltqp => /prime_oddPn-> //; rewrite -leqNgt prime_gt1. Qed. (* This is the second assertion of B & G, Appendix C, Remark V. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
odd_p
| |
odd_q : odd q. Proof. apply: contraR not_dvd_q_p1 => /prime_oddPn-> //. by rewrite -subn1 dvdn2 oddB ?odd_p. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
odd_q
| |
qgt2 : (2 < q)%N. Proof. by rewrite odd_prime_gt2. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
qgt2
| |
pgt4 : (4 < p)%N. Proof. by rewrite odd_geq ?(leq_ltn_trans qgt2). Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
pgt4
| |
qgt1 : (1 < q)%N. Proof. exact: ltnW. Qed. Local Notation Nm := (galNorm Fp Fpq). Local Notation uval := (@FinRing.uval _).
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
qgt1
| |
cycFU (FU : {group {unit F}}) : cyclic FU. Proof. exact: field_unit_group_cyclic. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
cycFU
| |
cUU : abelian U. Proof. by rewrite cyclic_abelian // -(injm_cyclic inj_sigmaU) ?cycFU. Qed. (* This is B & G, Appendix C, Remark VII. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
cUU
| |
im_psi (x : F) : (x \in psi @: U) = (Nm x == 1). Proof. have /cyclicP[u0 defFU]: cyclic [set: {unit F}] by apply: cycFU. have o_u0: #[u0] = (p ^ q).-1 by rewrite orderE -defFU card_finField_unit oF. have ->: psi @: U = uval @: (sigmaU @* U) by rewrite morphimEdom -imset_comp. have /set1P[->]: (sigmaU @* U)%G \in [set <[u0 ^+ (#[u0] %/ nU)]>%G]. rewrite -cycle_sub_group ?inE; last first. by rewrite o_u0 -(divnK (dvdn_pred_predX p q)) dvdn_mulr. by rewrite -defFU subsetT card_injm //= oU. rewrite divnA ?dvdn_pred_predX // -o_u0 mulKn //. have [/= alpha alpha_gen Dalpha] := finField_galois_generator (subvf Fp). have{} Dalpha x1: x1 != 0 -> x1 / alpha x1 = x1^-1 ^+ p.-1. move=> nz_x1; rewrite -[_ ^+ _](mulVKf nz_x1) -exprS Dalpha ?memvf // exprVn. by rewrite dimv1 oF_p prednK ?prime_gt0. apply/idP/(Hilbert's_theorem_90 alpha_gen (memvf _)) => [|[u [_ nz_u] ->]]. case/imsetP=> /= _ /cycleP[n ->] ->; rewrite expgAC; set u := (u0 ^+ n)%g. have nz_u: (val u)^-1 != 0 by rewrite -unitfE unitrV (valP u). by exists (val u)^-1; rewrite ?memvf ?Dalpha //= invrK val_unitX. have /cycleP[n Du]: (insubd u0 u)^-1%g \in <[u0]> by rewrite -defFU inE. have{} Du: u^-1 = val (u0 ^+ n)%g by rewrite -Du /= insubdK ?unitfE. by rewrite Dalpha // Du -val_unitX imset_f // expgAC mem_cycle. Qed. (* This is B & G, Appendix C, Remark VIII. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
im_psi
| |
defFU : sigmaU @* U \x [set u | uval u \in Fp] = [set: {unit F}]. Proof. have fP v: in_alg F (uval v) \is a GRing.unit by rewrite rmorph_unit ?(valP v). pose f (v : {unit 'F_p}) := FinRing.unit F (fP v). have fM: {in setT &, {morph f: v1 v2 / (v1 * v2)%g}}. by move=> v1 v2 _ _; apply: val_inj; rewrite /= -1?in_algE rmorphM. pose galFpU := Morphism fM @* [set: {unit 'F_p}]. have ->: [set u | uval u \in Fp] = galFpU. apply/setP=> u; rewrite inE /galFpU morphimEdom. apply/idP/imsetP=> [|[v _ ->]]; last by rewrite /= rpredZ // memv_line. case/vlineP=> v Du; have nz_v: v != 0. by apply: contraTneq (valP u) => v0; rewrite unitfE /= Du v0 scale0r eqxx. exists (insubd (1%g : {unit 'F_p}) v); rewrite ?inE //. by apply: val_inj; rewrite /= insubdK ?unitfE. have oFpU: #|galFpU| = p.-1. rewrite card_injm ?card_finField_unit ?oF_p //. by apply/injmP=> v1 v2 _ _ []/(fmorph_inj (in_alg F))/val_inj. have oUU: #|sigmaU @* U| = nU by rewrite card_injm. rewrite dprodE ?coprime_TIg ?oUU ?oFpU //; last first. by rewrite (sub_abelian_cent2 (cyclic_abelian (cycFU [set: _]))) ?subsetT. apply/eqP; rewrite eqEcard subsetT coprime_cardMg oUU oFpU //=. by rewrite card_finField_unit oF divnK ?dvdn_pred_predX. Qed. (* This is B & G, Appendix C, Remark IX. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
defFU
| |
frobH : [Frobenius H = P ><| U]. Proof. apply/Frobenius_semiregularP=> // [||u /setD1P[ntu Uu]]. - by rewrite -(morphim_injm_eq1 inj_sigma) // im_sigma finRing_nontrivial. - rewrite -cardG_gt1 oU ltn_divRL ?dvdn_pred_predX // mul1n -!subn1. by rewrite ltn_sub2r ?(ltn_exp2l 0) ?(ltn_exp2l 1) ?prime_gt1. apply/trivgP/subsetP=> x /setIP[Px /cent1P/commgP]. rewrite inE -!(morph_injm_eq1 inj_sigma) ?(sigmaE, in_PU) //. rewrite -mulrN1 addrC -mulrDr mulf_eq0 subr_eq0 => /orP[] // /idPn[]. by rewrite (inj_eq val_inj (sigmaU u) 1%g) morph_injm_eq1. Qed. (* From the abelQ assumption of Peterfalvi, Theorem (14.2) to the assumptions *) (* of part (B) of the assumptions of Theorem C. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
frobH
| |
p'q : q != p. Proof. by rewrite ltn_eqF. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
p'q
| |
cQQ : abelian Q. Proof. exact: abelem_abelian abelQ. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
cQQ
| |
p'Q : p^'.-group Q. Proof. exact: pi_pgroup (abelem_pgroup abelQ) _. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
p'Q
| |
pP : p.-group P. Proof. by rewrite /pgroup oP pnatX ?pnat_id. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
pP
| |
coQP : coprime #|Q| #|P|. Proof. exact: p'nat_coprime p'Q pP. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
coQP
| |
sQP0Q : [~: Q, P0] \subset Q. Proof. by rewrite commg_subl. Qed. (* This is B & G, Appendix C, Remark X. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
sQP0Q
| |
defQ : 'C_Q(P0) \x [~: Q, P0] = Q. Proof. by rewrite dprodC coprime_abelian_cent_dprod // (coprimegS sP0P). Qed. (* This is B & G, Appendix C, Remark XI. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
defQ
| |
nU_P0QP0 : exists2 y, y \in [~: Q, P0] & P0 :^ y \subset 'N(U). Proof. have [_ /(mem_dprod defQ)[z [y [/setIP[_ cP0z] QP0y -> _]]]] := nU_P0Q. by rewrite conjsgM (normsP (cent_sub P0)) //; exists y. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
nU_P0QP0
| |
E := [set x : galF | Nm x == 1 & Nm (2 - x) == 1].
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
E
| |
E_1 : 1 \in E. Proof. by rewrite !inE -addrA subrr addr0 galNorm1 eqxx. Qed. (* This is B & G, Appendix C, Lemma C.1. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
E_1
| |
Einv_gt1_le_pq : E = [set x^-1 | x in E] -> (1 < #|E|)%N -> (p <= q)%N. Proof. rewrite (cardsD1 1) E_1 ltnS card_gt0 => Einv /set0Pn[/= a /setD1P[not_a1 Ea]]. pose tau (x : F) := (2 - x)^-1. have Etau x: x \in E -> tau x \in E. rewrite inE => Ex; rewrite Einv (imset_f (fun y => y^-1)) //. by rewrite inE andbC opprD addNKr opprK. pose Pa := \prod_(beta in 'Gal(Fpq / Fp)) (beta (1 - a) *: 'X + 1). have galPoly_roots: all (root (Pa - 1)) (enum Fp). apply/allP=> x; rewrite mem_enum => /vlineP[b ->]. rewrite rootE !hornerE horner_prod subr_eq0 /=; apply/eqP. pose h k := (1 - a) *+ k + 1; transitivity (Nm (h b)). apply: eq_bigr => beta _. rewrite rmorphB rmorphD rmorphMn/= rmorphB /= rmorph1 /=. by rewrite -mulr_natr -scaler_nat natr_Zp hornerD hornerZ hornerX hornerC. elim: (b : nat) => [|k IHk]; first by rewrite /h add0r galNorm1. suffices{IHk}: h k / h k.+1 \in E. rewrite inE -invr_eq1 => /andP[/eqP <- _]. by rewrite galNormM galNormV /= [galNorm _ _ (h k)]IHk mul1r invrK. elim: k => [|k IHk]; first by rewrite /h add0r mul1r addrAC Etau. have nz_hk1: h k.+1 != 0. apply: contraTneq IHk => ->; rewrite invr0 mulr0. by rewrite inE galNorm0 eq_sym oner_eq0. congr (_ \in E): (Etau _ IHk); apply: canLR (@invrK _) _; rewrite invfM invrK. apply: canRL (mulKf nz_hk1) _; rewrite mulrC mulrBl divfK // mulrDl mul1r. by rewrite {2}/h mulrS -(addrA (1 -a)) (addrA _ (1 -a)) addrK addrAC -mulrSr. have sizePa: size Pa = q.+1. have sizePaX (beta : {rmorphism F -> F}) : size (beta (1 - a) *: 'X + 1) = 2%N. rewrite -mul_polyC size_MXaddC oner_eq0 andbF size_polyC fmorph_eq0. by rewrite subr_eq0 eq_sym (negbTE not_a1). rewrite size_prod => [|i _]; last by rewrite -size_poly_eq0 sizePaX. rewrite (eq_bigr (fun _ => 2%N)) => [|beta _]; last by rewrite sizePaX. rewrite sum_nat_const muln2 -addnn -addSn addnK. by rewrite -galois_dim ?finField_galois ?subvf // dimv1 divn1 dimFpq. have sizePa1: size (Pa - 1) = q.+1. by rewrite size_addl // size_opp size_poly1 sizePa. have nz_Pa1 : Pa - 1 != 0 by rewrite -size_poly_eq0 sizePa1. by rewrite -ltnS -oFp -sizePa1 cardE max_poly_roots ?enum_uniq. Qed. (* This is B & G, Appendix C, Lemma C.2. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
Einv_gt1_le_pq
| |
E_gt1 : (1 < #|E|)%N. Proof. have [q_gt4 | q_le4] := ltnP 4 q. pose inK x := enum_rank_in (classes1 H) (x ^: H). have inK_E x: x \in H -> enum_val (inK x) = x ^: H. by move=> Hx; rewrite enum_rankK_in ?mem_classes. pose j := inK s; pose k := inK (s ^+ 2)%g; pose e := gring_classM_coef j j k. have cPP: abelian P by rewrite -(injm_abelian inj_sigma) ?zmod_abelian. have Hs: s \in H by rewrite -(sdprodW defH) -[s]mulg1 mem_mulg. have DsH n: (s ^+ n) ^: H = (s ^+ n) ^: U. rewrite -(sdprodW defH) classM (abelian_classP _ cPP) ?groupX //. by rewrite class_support_set1l. have injJU: {in U &, injective (conjg s)}. by move=> u v Uu Uv eq_s_uv; apply/inj_psi; rewrite ?psiE ?eq_s_uv. have ->: #|E| = e. rewrite /e /gring_classM_coef !inK_E ?groupX //. transitivity #|[set u in U | s^-1 ^ u * s ^+ 2 \in s ^: U]%g|. rewrite -(card_in_imset (sub_in2 _ inj_psi)) => [|u /setIdP[] //]. apply: eq_card => x; rewrite inE -!im_psi. apply/andP/imsetP=> [[/imsetP[u Uu ->] /imsetP[v Uv Dv]]{x} | ]. exists u; rewrite // inE Uu /=; apply/imsetP; exists v => //. by apply: (injmP inj_sigma); rewrite ?(sigmaE, in_PU) // mulN1r addrC. case=> u /setIdP[Uu /imsetP[v Uv /(congr1 sigma)]]. rewrite ?(sigmaE, in_PU) // mulN1r addrC => Dv ->. by rewrite Dv !imset_f. rewrite DsH (DsH 1%N) expg1; have [w Uw ->] := repr_class U (s ^+ 2). pose f u := (s ^ (u * w), (s^-1 ^ u * s ^+ 2) ^ w). rewrite -(@card_in_imset _ _ f) => [|u v]; last first. by move=> /setIdP[Uu _] /setIdP[Uv _] [/injJU/mulIg-> //]; apply: groupM. apply: eq_card => [[x1 x2]]; rewrite inE -andbA. apply/imsetP/and3P=> [[u /setIdP[Uu sUs2u'] [-> ->]{x1 x2}] | []]. rewrite /= conjgM -(rcoset_id Uw) class_rcoset !memJ_conjg mem_orbit //. by rewrite sUs2u' -conjMg conjVg mulKVg. case/imsetP=> u Uu /= -> sUx2 /eqP/(canRL (mulKg _)) Dx2. exists (u * w^-1)%g; last first. by rewrite /f /= conjMg -conjgM mulgKV conjVg -Dx2. rewrite inE !in_PU // Uw -(memJ_conjg _ w) -class_rcoset rcoset_id //. by rewrite conjMg -conjgM mulgKV conjVg -Dx2. pose chi_s2 i := 'chi[H]_i s ^+ 2 * ('chi_i (s ^+ 2)%g)^* / 'chi_i 1%g. have De: e%:R = #|U|%:R / #|P|%:R * (\sum_i chi_s2 i). have Ks: s \in enum_val j by rewrite inK_E ?class_refl. have Ks2: (s ^+ 2)%g \in enum_val k by rewrite inK_E ?groupX ?class_refl. rewrite (gring_classM_coef_sum_eq Ks Ks Ks2) inK_E //; congr (_ * _). have ->: #|s ^: H| = #|U| by rewrite (DsH 1%N) (card_in_imset injJU). by rewrite -(sdprod_card defH) mulnC !natrM invfM mulrA mulfK ?neq0CG. pose linH := [pred i | P \subset cfker 'chi[H]_i]. have nsPH: P <| H by have [] := sdprod_context defH. have sum_linH: \sum_(i in linH) chi_s2 i = #|U|%:R. have isoU: U \isog H / P := sdprod_isog defH. have abHbar: abelian (H / P) by rewrite -(isog_abelian isoU). rewrite (card_isog isoU) -(card_Iirr_abelian abHbar) -sumr_const. rewrite (reindex _ (mod_Iirr_bij nsPH)) /chi_s2 /=. apply: eq_big => [i | i _]; rewrite ?mod_IirrE ?cfker_mod //. have lin_i: ('chi_i %% P)%CF \is a linear_char. exact/cfMod_lin_char/char_abelianP. rewrite lin_char1 // divr1 -lin_charX // -normCK. by rewrite normC_lin_char ?groupX ?expr1n. have degU i: i \notin linH -> 'chi_i 1%g = #|U|%:R. case/(Frobenius_Ind_irrP (FrobeniusWker frobH)) => {}i _ ->. rewrite cfInd1 ?normal_sub // -(index_sdprod defH) lin_char1 ?mulr1 //. exact/char_abelianP. have ub_linH' m (s_m := (s ^+ m)%g): (0 < m < 5)%N -> \sum_(i in predC linH) `|'chi_i s_m| ^+ 2 <= #|P|%:R. - case/andP=> m_gt0 m_lt5; have{m_gt0 m_lt5} P1sm: s_m \in P^#. rewrite !inE groupX // -order_dvdn -(order_injm inj_sigma) // sigmaE. by rewrite andbT order_primeChar ?oner_neq0 ?gtnNdvd ?(leq_trans m_lt5). have ->: #|P| = (#|P| * (s_m \in s_m ^: H))%N by rewrite class_refl ?muln1. have{P1sm} /eqP <-: 'C_H[s ^+ m] == P. rewrite eqEsubset (Frobenius_cent1_ker frobH) // subsetI normal_sub //=. by rewrite sub_cent1 groupX // (subsetP cPP). rewrite mulrnA -second_orthogonality_relation ?groupX // big_mkcond. by apply: ler_sum => i _; rewrite normCK; case: ifP; rewrite ?mul_conjC_ge0. have sqrtP_gt0: 0 < sqrtC #|P|%:R :> algC by rewrite sqrtC_gt0 ?gt0CG. have{De ub_linH'}: `|(#|P| * e)%:R - #|U|%:R ^+ 2 : algC| <= #|P|%:R * sqrtC #|P|%:R. rewrite natrM De mulrCA mulrA divfK ?neq0CG // (bigID linH) /= sum_linH. rewrite mulrDr addrC addKr mulrC mulr_suml /chi_s2. rewrite (le_trans (ler_norm_sum _ _ _)) // -ler_pdivrMr // mulr_suml. apply: le_trans (ub_linH' 1%N isT); apply: ler_sum => i linH'i. rewrite ler_pdivrMr // degU ?divfK ?neq0CG //. rewrite normrM -normrX norm_conjC ler_wpM2l ?normr_ge0 //. rewrite -ler_sqr ?qualifE /= ?normr_ge0 ?(ltW (x := 0)) // ?sqrtCK. apply: le_trans (ub_linH' 2%N isT); rewrite (bigD1 i) ?ler_wpDr //=. by apply: sumr_ge0 => i1 _; rewrite exprn_ge0 ?normr_ge0. rewrite natrM real_ler_distl ?rpredB ?rpredM ?rpred_nat // => /andP[lb_Pe _]. rewrite -ltC_nat -(ltr_pM2l (gt0CG P)) {lb_Pe}(lt_le_trans _ lb_Pe) //. rewrite ltrBrDl (le_lt_trans (y := (p ^ q.-1)%:R ^+ 2)) //; last first. rewrite -!natrX ltC_nat ltn_sqr oU ltn_divRL ?dvdn_pred_predX //. rewrite -(subnKC qgt1) /= -!subn1 mulnBr muln1 -expnSr. by rewrite ltn_sub2l ?(ltn_exp2l 0) // prime_gt1. rewrite -mulrDr -natrX -expnM muln2 -subn1 doubleB -addnn -addnBA // subn2. rewrite expnD natrM -oP ler_wpM2l ?ler0n //. apply: le_trans (_ : 2 * sqrtC #|P|%:R <= _). rewrite mulrDl mul1r lerD2l -(@expr_ge1 _ 2) ?(ltW sqrtP_gt0) // sqrtCK. by rewrite oP natrX expr_ge1 ?ler0n ?ler1n. rewrite -ler_sqr ?rpredM ?rpred_nat ?qualifE ?(ltW sqrtP_gt0) //. rewrite exprMn sqrtCK -!natrX -natrM leC_nat -expnM muln2 oP. rewrite -(subnKC q_gt4) doubleS (expnS p _.*2.+1) -(subnKC pgt4) leq_mul //. by rewrite ?leq_exp2l // !doubleS !ltnS -addnn leq_addl. have q3: q = 3%N by apply/eqP; rewrite eqn_leq qgt2 andbT -ltnS -(odd_ltn 5). rewrite (cardsD1 1) E_1 ltnS card_gt0; apply/set0Pn => /=. pose f (c : 'F_p) : {poly 'F_p} := 'X * ('X - 2%:P) * ('X - c%:P) + ('X - 1). (* TODO when requiring mathcomp >= 2.4.0, the following three lines can be simplified to have fc0 c: (f c).[0] = -1 by rewrite !hornerE. have fc2 c: (f c).[2] = 1 by rewrite !(subrr, hornerE) /= addrK. c.f. https://github.com/math-comp/odd-order/pull/68#discussion_r2030878922 *) have fc0 c: (f c).[0] = -1 by rewrite !hornerE //= !hornerE. have fc2 c: (f c).[2] = 1 by rewrite !(subrr, hornerE) /= addrK; apply/val_inj. have /existsP[c nz_fc]: [exists c, ~~ [exists d, root (f c) d]]. have nz_f_0 c: ~~ root (f c) 0 by rewrite /root fc0 oppr_eq0. rewrite -negb_forall; apply/negP=> /'forall_existsP/fin_all_exists[/= rf rfP]. suffices inj_rf: injective rf. by have /negP[] := nz_f_0 (invF inj_rf 0); rewrite -{2}[0](f_invF inj_rf). move=> a b eq_rf_ab; apply/oppr_inj/(addrI (rf a)). have: (f a).[rf a] = (f b).[rf a] by rewrite {2}eq_rf_ab !(rootP _). rewrite !(hornerXsubC, hornerD, hornerM) hornerX => /addIr/mulfI-> //. rewrite mulf_neq0 ?subr_eq0 1?(contraTneq _ (rfP a)) // => -> //. by rewrite /root fc2. have{nz_fc} /= nz_fc: ~~ root (f c) _ by apply/forallP; rewrite -negb_exists. have sz_fc_lhs: size ('X * ('X - 2%:P) * ('X - c%:P)) = 4%N. by rewrite !(size_mul, =^~ size_poly_eq0) ?size_polyX ?size_XsubC. have sz_fc: size (f c) = 4%N by rewrite size_addl ?size_XsubC sz_fc_lhs. have irr_fc: irreducible_poly (f c) by apply: cubic_irreducible; rewrite ?sz_fc. have fc_monic : f c \is monic. rewrite monicE lead_coefDl ?size_XsubC ?sz_fc_lhs // -monicE. by rewrite !monicMl ?monicXsubC ?monicX. pose inF : {rmorphism _ -> _} := in_alg F; pose fcF := map_poly inF (f c). have /existsP[a fcFa_0]: [exists a : F, root fcF a]. suffices: ~~ coprimep (f c) ('X ^+ #|F| - 'X). apply: contraR; rewrite -(coprimep_map inF) negb_exists => /forallP-nz_fcF. rewrite -/fcF rmorphB rmorphXn /= map_polyX finField_genPoly. elim/big_rec: _ => [|x gF _ co_fcFg]; first exact: coprimep1. by rewrite coprimepMr coprimep_XsubC nz_fcF. have /irredp_FAdjoin[L dimL [z /coprimep_root fcz0 _]] := irr_fc. pose finL := Vector.clone 'F_p (FinFieldExtType L) _. set fcL := map_poly _ _ in fcz0; pose inL : {rmorphism _ -> _} := in_alg L. rewrite -(coprimep_map inL) -/fcL rmorphB rmorphXn /= map_polyX. apply: contraL (fcz0 _) _; rewrite hornerD hornerN hornerXn hornerX subr_eq0. have ->: #|F| = #|{: finL}%VS| by rewrite oF card_vspace dimL sz_fc oF_p q3. by rewrite card_vspacef (expf_card (z : finL)). have Fp_fcF: fcF \is a polyOver Fp. by apply/polyOverP => i; rewrite coef_map /= memvZ ?memv_line. pose G := 'Gal(Fpq / Fp). have galG: galois Fp Fpq by rewrite finField_galois ?subvf. have oG: #|G| = 3%N by rewrite -galois_dim // dimv1 dimFpq q3. have Fp'a: a \notin Fp. by apply: contraL fcFa_0 => /vlineP[d ->]; rewrite fmorph_root. have DfcF: fcF = \prod_(beta in G) ('X - (beta a)%:P). pose Pa : {poly F} := minPoly Fp a. have /eqP szPa: size Pa == 4%N. rewrite size_minPoly eqSS. rewrite (sameP eqP (prime_nt_dvdP _ _)) ?adjoin_deg_eq1 //. by rewrite adjoin_degreeE dimv1 divn1 -q3 -dimFpq field_dimS ?subvf. have dvd_Pa_fcF: Pa %| fcF by apply: minPoly_dvdp fcFa_0. have{dvd_Pa_fcF} /eqP <-: Pa == fcF. rewrite -eqp_monic ?monic_minPoly ?monic_map // -dvdp_size_eqp //. by rewrite szPa size_map_poly sz_fc. have [r [srG /map_uniq Ur defPa]]:= galois_factors (subvf _) galG a (memvf a). rewrite -/Pa big_map in defPa; rewrite defPa big_uniq //=. apply/eq_bigl/subset_cardP=> //; apply/eqP. by rewrite -eqSS (card_uniqP Ur) oG -szPa defPa size_prod_XsubC. exists a; rewrite !inE; apply/and3P; split. - by apply: contraNneq Fp'a => ->; apply: mem1v. - apply/eqP; transitivity ((- 1) ^+ #|G| * fcF.[inF 0]). rewrite DfcF horner_prod -prodrN; apply: eq_bigr => beta _. by rewrite rmorph0 hornerXsubC add0r opprK. by rewrite -signr_odd mulr_sign oG horner_map fc0 rmorphN1 opprK. apply/eqP; transitivity (fcF.[inF 2]); last by rewrite horner_map fc2 rmorph1. rewrite DfcF horner_prod; apply: eq_bigr => beta _. by rewrite hornerXsubC rmorphB !rmorph_nat. Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
E_gt1
| |
Qy : y \in Q. Proof. by rewrite (subsetP sQP0Q). Qed.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
Qy
| |
t := s ^ y.
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
t
| |
P1 := P0 :^ y. (* This is B & G, Appendix C, Lemma C.3, Step 1. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
P1
| |
splitH x : x \in H -> exists2 u, u \in U & exists2 v, v \in U & exists2 s1, s1 \in P0 & x = u * s1 * v. Proof. case/(mem_sdprod defH) => z [v [Pz Uv -> _]]. have [-> | nt_z] := eqVneq z 1. by exists 1 => //; exists v => //; exists 1; rewrite ?mulg1. have nz_z: sigma z != 0 by rewrite (morph_injm_eq1 inj_sigma). have /(mem_dprod defFU)[]: finField_unit nz_z \in setT := in_setT _. move=> _ [w [/morphimP[u Uu _ ->] Fp_w /(congr1 val)/= Dz _]]. have{Fp_w Dz} [n Dz]: exists n, sigma z = sigma ((s ^+ n) ^ u). move: Fp_w; rewrite {}Dz inE => /vlineP[n ->]; exists n. by rewrite -{1}(natr_Zp n) scaler_nat mulr_natr conjXg !sigmaE ?in_PU. exists u^-1; last exists (u * v); rewrite ?groupV ?groupM //. exists (s ^+ n); rewrite ?groupX // mulgA; congr (_ * _). by apply: (injmP inj_sigma); rewrite // -?mulgA // -conjgE ?in_PU. Qed. (* This is B & G, Appendix C, Lemma C.3, Step 2. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
splitH
| |
not_splitU s1 s2 u : s1 \in P0 -> s2 \in P0 -> u \in U -> s1 * u * s2 \in U -> (s1 == 1) && (s2 == 1) || (u == 1) && (s1 * s2 == 1). Proof. move=> P0s1 P0s2 Uu; have [_ _ _ tiPU] := sdprodP defH. have [Ps1 Ps2]: s1 \in P /\ s2 \in P by rewrite !(subsetP sP0P). have [-> | nt_s1 /=] := altP (s1 =P 1). by rewrite mul1g groupMl // -in_set1 -set1gE -tiPU inE Ps2 => ->. have [-> | nt_u /=] := altP (u =P 1). by rewrite mulg1 -in_set1 -set1gE -tiPU inE (groupM Ps1). rewrite (conjgC _ u) -mulgA groupMl // => Us12; case/negP: nt_u. rewrite -(morph_injm_eq1 inj_sigmaU) // -in_set1 -set1gE. have [_ _ _ <-] := dprodP defFU; rewrite !inE mem_morphim //= -/(psi u). have{Us12}: s1 ^ u * s2 == 1. by rewrite -in_set1 -set1gE -tiPU inE Us12 andbT !in_PU. rewrite -(morph_injm_eq1 inj_sigma) ?(in_PU, sigmaE) // addr_eq0. move/eqP/(canRL (mulKf _))->; rewrite ?morph_injm_eq1 //. by rewrite mulrC rpred_div ?rpredN //= -sigmaP0 mem_morphim. Qed. (* This is B & G, Appendix C, Lemma C.3, Step 3. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
not_splitU
| |
tiH_P1 t1 : t1 \in P1^# -> H :&: H :^ t1 = U. Proof. case/setD1P=>[nt_t1 P1t1]; set X := H :&: _. have [nsPH sUH _ _ tiPU] := sdprod_context defH. have sUX: U \subset X. by rewrite subsetI sUH -(normsP nUP0y t1 P1t1) conjSg. have defX: (P :&: X) * U = X. by rewrite setIC group_modr // (sdprodW defH) setIAC setIid. have [tiPX | ntPX] := eqVneq (P :&: X) 1; first by rewrite -defX tiPX mul1g. have irrPU: acts_irreducibly U P 'J. apply/mingroupP; (split=> [|V /andP[ntV]]; rewrite astabsJ) => [|nVU sVP]. by have [_ ->] := Frobenius_context frobH. apply/eqP; rewrite eqEsubset sVP; apply/subsetP=> x Px. have [-> // | ntx] := eqVneq x 1. have [z Vz ntz] := trivgPn _ ntV; have Pz := subsetP sVP z Vz. have nz_z: sigma z != 0%R by rewrite morph_injm_eq1. have uP: (sigma x / sigma z)%R \is a GRing.unit. by rewrite unitfE mulf_neq0 ?invr_eq0 ?morph_injm_eq1. have: FinRing.unit F uP \in setT := in_setT _. case/(mem_dprod defFU)=> _ [s1 [/morphimP[u Uu _ ->]]]. rewrite inE => /vlineP[n Ds1] /(congr1 val)/= Dx _. suffices ->: x = (z ^ u) ^+ n by rewrite groupX ?memJ_norm ?(subsetP nVU). apply: (injmP inj_sigma); rewrite ?(in_PU, sigmaE) //. by rewrite -mulr_natr -scaler_nat natr_Zp -Ds1 -mulrA -Dx mulrC divfK. have{ntPX irrPU} defX: X :=: H. rewrite -(sdprodW defH) -defX; congr (_ * _). have [_ -> //] := mingroupP irrPU; rewrite ?subsetIl //= -/X astabsJ ntPX. by rewrite normsI // normsG. have nHt1: t1 \in 'N(H) by rewrite -groupV inE sub_conjgV; apply/setIidPl. have oP0: #|P0| = p by rewrite -(card_injm inj_sigma) // (eq_card sigmaP0) oFp. have{nHt1} nHP1: P1 \subset 'N(H). apply: prime_meetG; first by rewrite cardJg oP0. by apply/trivgPn; exists t1; rewrite // inE P1t1. have{nHP1} nPP1: P1 \subset 'N(P). have /Hall_pi hallP: Hall H P by apply: Frobenius_ker_Hall frobH. by rewrite -(normal_Hall_pcore hallP nsPH) gFnorm_trans. have sylP0: p.-Sylow(Q <*> P0) P0. rewrite /pHall -divgS joing_subr ?(pgroupS sP0P) //=. by rewrite norm_joinEr // coprime_cardMg ?(coprimegS sP0P) ?mulnK. have sP1QP0: P1 \subset Q <*> P0. by rewrite conj_subG ?joing_subr ?mem_gen // inE Qy. have nP10: P1 \subset 'N(P0). have: P1 \subset 'N(P :&: (Q <*> P0)) by rewrite normsI // normsG. by rewrite norm_joinEr // -group_modr // setIC coprime_TIg // mul1g. have eqP10: P1 :=: P0. apply/eqP; rewrite eqEcard cardJg leqnn andbT. rewrite (comm_sub_max_pgroup (Hall_max sylP0)) //; last exact: normC. by rewrite pgroupJ (pHall_pgroup sylP0). have /idPn[] := prime_gt1 pr_p. rewrite -oP0 cardG_gt1 negbK -subG1 -(Frobenius_trivg_cent frobH) subsetI sP0P. apply/commG1P/trivgP; rewrite -tiPU commg_subI // subsetI ?subxx //. by rewrite sP0P -eqP10. Qed. (* This is B & G, Appendix C, Lemma C.3, Step 4. *)
|
Let
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
tiH_P1
| |
BGappendixC3_Ediv : E = [set x^-1 | x in E]%R. Proof. suffices sEV_E: [set x^-1 | x in E]%R \subset E. by apply/esym/eqP; rewrite eqEcard sEV_E card_imset //=; apply: invr_inj. have /mulG_sub[/(subset_trans sP0P)/subsetP-sP0H /subsetP-sUH] := sdprodW defH. have Hs := sP0H s P0s; have P1t: t \in P1 by rewrite memJ_conjg. have nUP1 t1: t1 \in P1 -> U :^ t1 = U by move/(subsetP nUP0y)/normP. have nUtn n u: u \in U -> u ^ (t ^+ n) \in U. by rewrite -mem_conjgV nUP1 ?groupV ?groupX. have nUtVn n u: u \in U -> u ^ (t ^- n) \in U. by rewrite -mem_conjg nUP1 ?groupX. have Qsti i: s ^- i * t ^+ i \in Q. by rewrite -conjXg -commgEl (subsetP sQP0Q) // commGC mem_commg ?groupX. pose is_sUs m a j n u s1 v := [/\ a \in U, u \in U, v \in U, s1 \in P0 & s ^+ m * a ^ t ^+ j * s ^- n = u * s1 * v]. have split_sUs m a j n: a \in U -> exists u, exists s1, exists v, is_sUs m a j n u s1 v. - move=> Ua; suffices: s ^+ m * a ^ t ^+ j * s ^- n \in H. by case/splitH=> u Uu [v Uv [s1 P0s1 Dusv1]]; exists u, s1, v. by rewrite 2?groupM ?groupV ?groupX // sUH ?nUtn. have nt_sUs m j n a u s1 v: (m == n.+1) || (n == m.+1) -> is_sUs m a j n u s1 v -> s1 != 1. - move/pred2P=> Dmn [Ua Uu Uv _ Dusv]; have{Dmn}: s ^+ m != s ^+ n. by case: Dmn => ->; last rewrite eq_sym; rewrite expgS eq_mulgV1 ?mulgK. apply: contraNneq => s1_1; rewrite {s1}s1_1 mulg1 in Dusv. have:= groupM Uu Uv; rewrite -Dusv => /(not_splitU _ _ (nUtn j a Ua))/orP. by rewrite !in_group // eq_invg1 -eq_mulgV1 => -[]// /andP[? /eqP->]. have sUs_modP m a j n u s1 v: is_sUs m a j n u s1 v -> a ^ t ^+ j = u * v. have [nUP /isom_inj/injmP/=quoUP_inj] := sdprod_isom defH. case=> Ua Uu Uv P0s1 /(congr1 (coset P)); rewrite (conjgCV u) -(mulgA _ u). rewrite coset_kerr ?groupV ?groupX //. rewrite coset_kerl ?groupX // [RHS]coset_kerl; last first. by rewrite -mem_conjg (normsP nUP) // (subsetP sP0P). by move/quoUP_inj->; rewrite ?nUtn ?groupM. have expUMp u v Uu Uv := expgMn p (centsP cUU u v Uu Uv). have sUsXp m a j n u s1 v: is_sUs m a j n u s1 v -> is_sUs m (a ^+ p) j n (u ^+ p) s1 (v ^+ p). - move=> Dusv; have{Dusv} [/sUs_modP Duv [Ua Uu Vv P0s1 Dusv]] := (Dusv, Dusv). split; rewrite ?groupX //; move: P0s1 Dusv; rewrite -defP0 => /cycleP[k ->]. rewrite conjXg -!(mulgA _ (s ^+ k)) ![s ^+ k * _]conjgC [RHS]mulgA. rewrite (mulgA (u ^+ p)) -expUMp //. rewrite {}Duv ![s ^+ m * _]conjgC !conjXg -![_ * _ * s ^- n]mulgA. move/mulgI/(congr1 (Frobenius_aut charFp \o sigma))=> /= Duv_p. congr (_ * _); apply/(injmP inj_sigma); rewrite ?in_PU //. by rewrite !{1}sigmaE ?in_PU // rmorphB !rmorphMn rmorph1 in Duv_p *. have odd_P: odd #|P| by rewrite oP oddX odd_p orbT. suffices EpsiV a: a \in U -> psi a \in E -> psi (a^-1 ^ t ^+ 3) \in E. apply/subsetP => _ /imsetP[x Ex ->]. have /imsetP[a Ua Dx]: x \in psi @: U by rewrite im_psi; case/setIdP: Ex. suffices: psi (a^-1 ^ t ^+ (3 * #|P|)) \in E. rewrite Dx -psiV // -{2}(conjg1 a^-1); congr (psi (_ ^ _) \in E). by apply/eqP; rewrite -order_dvdn orderJ dvdn_mull ?order_dvdG. rewrite -(odd_double_half #|P|) odd_P addnC. elim: _./2 => [|n /EpsiV/EpsiV/=]; first by rewrite EpsiV -?Dx. by rewrite conjVg invgK -!conjgM -!expgD -!mulnSr !(groupV, nUtn) //; apply. move=> Ua Ea; have{Ea} [b Ub Dab]: exists2 b, b \in U & psi a + psi b = 2. case/setIdP: Ea => _; rewrite -im_psi => /imsetP[b Ub Db]; exists b => //. by rewrite -Db addrC subrK. (* In the book k is arbitrary in Fp; however only k := 3 is used. *) have [u2 [s2 [v2 usv2P]]] := split_sUs 3%N (a * _) 2%N 1%N (groupM Ua (groupVr Ub)). have{Ua} [u1 [s1 [v1 usv1P]]] := split_sUs 1%N a^-1 3%N 2%N (groupVr Ua). have{Ub} [u3 [s3 [v3 usv3P]]] := split_sUs 2%N b 1%N 3%N Ub. pose s2def w1 w2 w3 := t * s2^-1 * t = w1 * s3 * w2 * t ^+ 2 * s1 * w3. pose w1 := v2 ^ t^-1 * u3; pose w2 := v3 * u1 ^ t ^- 2; pose w3 := v1 * u2 ^ t. have stXC m n: (m <= n)%N -> s ^- n ^ t ^+ m = s ^- m ^ t ^+ n * s ^- (n - m). move/subnK=> Dn; apply/(mulgI (s ^- (n - m) * t ^+ n))/(mulIg (t ^+ (n - m))). rewrite -{1}[in t ^+ n]Dn expgD !mulgA !mulgK -invMg -3![LHS]mulgA -!expgD. by rewrite addnC Dn mulgA (centsP (abelem_abelian abelQ)) ?mulgA. wlog suffices Ds2: a b u1 v1 u2 v2 u3 v3 @w1 @w2 @w3 Dab usv1P usv2P usv3P / s2def w1 w2 w3; last first. - apply/esym; rewrite -[_ * t]mulgA [_ * t]conjgC [RHS]mulgA -(expgS _ 1) conjVg. rewrite /w2 (mulgA _ v3); apply: (canRL (mulKVg _)). rewrite 2!(mulgA (t ^- 2)) -conjgE. rewrite conjMg conjgKV /w3 mulgA; apply: (canLR (mulgKV _)). rewrite /w1 -2!mulgA -(mulgA _ s3) -(mulgA _ u3). rewrite (mulgA u1) (mulgA u3) conjMg -conjgM mulKg -[LHS]mulgA. have [[[Ua _ _ _ <-] [_ _ _ _ Ds2]] [Ub _ _ _ <-]] := (usv1P, usv2P, usv3P). apply: (canLR (mulKVg _)); rewrite -!invMg -!conjMg -{}Ds2 groupV in Ua *. rewrite -[t]expg1 2!conjMg -conjgM -expgS 2![in RHS]conjMg -conjgM -expgSr mulgA. apply: (canLR (mulgK _)); rewrite [in RHS]invMg (invMg (_ ^ _)). rewrite -!conjVg invgK (invMg a) invgK. rewrite -2!mulgA -[RHS]mulgA -[X in _ = _ * X]mulgA. rewrite (mulgA _ s) stXC // mulgKV -!conjMg stXC // mulgKV -conjMg conjgM. apply: (canLR (mulKVg _)); rewrite -2!conjVg 2!mulgA -conjMg (stXC 1%N) //. rewrite mulgKV -conjgM -expgSr -mulgA -!conjMg; congr (_ ^ t ^+ 3). apply/(canLR (mulKVg _))/(canLR (mulgK _))/(canLR invgK). rewrite -!mulgA (mulgA _ b) (mulgA b^-1) invMg -!conjVg !invgK. by apply/(injmP inj_sigma); rewrite 1?groupM ?sigmaE ?memJ_P. have [[Ua Uu1 Uv1 P0s1 Dusv1] /sUs_modP-Duv1] := (usv1P, usv1P). have [[_ Uu2 Uv2 P0s2 _] [Ub Uu3 Uv3 P0s3 _]] := (usv2P, usv3P). suffices /(congr1 sigma): s ^+ 2 = s ^ v1 * s ^ a^-1 ^ t ^+ 3. rewrite inE sigmaX // sigma_s sigmaM ?memJ_P -?psiE ?nUtn // => ->. by rewrite addrK -!im_psi !imset_f ?nUtn. rewrite groupV in Ua; have [Hs1 Hs3]: s1 \in H /\ s3 \in H by rewrite !sP0H. have nt_s1: s1 != 1 by apply: nt_sUs usv1P. have nt_s3: s3 != 1 by apply: nt_sUs usv3P. have{sUsXp} Ds2p: s2def (w1 ^+ p) (w2 ^+ p) (w3 ^+ p). have [/sUsXp-usv1pP /sUsXp-usv2pP /sUsXp-usv3pP] := And3 usv1P usv2P usv3P. rewrite expUMp ?groupV // !expgVn in usv1pP usv2pP. rewrite !(=^~ conjXg _ _ p, expUMp) ?groupV -1?[t]expg1 ?nUtn ?nUtVn //. apply: Ds2 usv1pP usv2pP usv3pP => //. by rewrite !psiX // -!Frobenius_autE -rmorphD Dab rmorph_nat. have{} Ds2: s2def w1 w2 w3 by apply: Ds2 usv1P usv2P usv3P. wlog [Uw1 Uw2 Uw3]: w1 w2 w3 Ds2p Ds2 / [/\ w1 \in U, w2 \in U & w3 \in U]. by move/(_ w1 w2 w3)->; rewrite ?(nUtVn, nUtVn 1%N, nUtn 1%N, in_group). have{Ds2p} Dw3p: (w2 ^- p * w1 ^- p.-1 ^ s3 * w2) ^ t ^+ 2 = w3 ^+ p.-1 ^ s1^-1. rewrite -[w1 ^+ _](mulKg w1) -[w3 ^+ _](mulgK w3) -expgS -expgSr !prednK //. rewrite -(canLR (mulKg _) Ds2p) -(canLR (mulKg _) Ds2). rewrite 3!invMg (invMg _ (w2 ^+ p)) (invMg _ s3) (invMg (_^-1)) !invgK. rewrite [X in _ = X ^ _](mulgA _ _^-1). by rewrite mulgK [2%N]lock /conjg !mulgA mulVg mul1g mulgK. have w_id w: w \in U -> w ^+ p.-1 == 1 -> w = 1. by move=> Uw /eqP/(canRL_in (expgK _) Uw)->; rewrite ?expg1n ?oU. have{Uw3} Dw3: w3 = 1. apply: w_id => //; have:= @not_splitU s1^-1^-1 s1^-1 (w3 ^+ p.-1). rewrite !groupV mulVg eqxx andbT {2}invgK (negPf nt_s1) groupX //= => -> //. have /tiH_P1 <-: t ^+ 2 \in P1^#. rewrite 2!inE groupX // andbT -order_dvdn gtnNdvd // orderJ. by rewrite odd_gt2 ?order_gt1 // orderE defP0 (oddSg sP0P). by rewrite -mulgA -conjgE inE -{2}Dw3p memJ_conjg !in_group ?Hs1 // sUH. have{Dw3p} Dw2p: w2 ^+ p.-1 = w1 ^- p.-1 ^ s3. apply/(mulIg w2)/eqP; rewrite -expgSr prednK // eq_mulVg1 mulgA. by rewrite (canRL (conjgK _) Dw3p) Dw3 expg1n !conj1g. have{Uw1} Dw1: w1 = 1. apply: w_id => //; have:= @not_splitU s3^-1 s3 (w1 ^- p.-1). rewrite mulVg (negPf nt_s3) andbF -mulgA -conjgE -Dw2p !in_group //=. by rewrite eqxx andbT eq_invg1 /= => ->. have{w1 w2 w3 Dw1 Dw3 w_id Uw2 Dw2p} Ds2: t * s2^-1 * t = s3 * t ^+ 2 * s1. by rewrite Ds2 Dw3 [w2]w_id ?mulg1 ?Dw2p ?Dw1 ?mul1g // expg1n invg1 conj1g. have /centsP abP0: abelian P0 by rewrite -defP0 cycle_abelian. have QP0ys := memJ_norm y (subsetP (commg_normr P0 Q) _ _). have{QP0ys} memQP0 := (QP0ys, groupV, groupM); have nQ_P0 := subsetP nQP0. have sQP0_Q: [~: Q, P0] \subset Q by rewrite commg_subl. have /centsP abQP0 := abelianS sQP0_Q (abelem_abelian abelQ). have{s2def} Ds312: s3 * s1 * s2 = 1. apply/set1P; rewrite -set1gE -(coprime_TIg coQP) inE. rewrite coset_idr ?(subsetP sP0P) ?nQ_P0 ?groupM //. rewrite -mulgA -[s2](mulgK s) [_ * s]abP0 // -[s2](mulKVg s). rewrite -!mulgA [s * _]mulgA [s1 * _]mulgA [s1 * _]abP0 ?groupM //. rewrite 2!(mulgA s3) [s^-1 * _]mulgA !(morphM, morphV) ?nQ_P0 ?in_group //=. have ->: coset Q s = coset Q t by rewrite coset_kerl ?groupV ?coset_kerr. have nQt: t \in 'N(Q) by rewrite -(conjGid Qy) normJ memJ_conjg nQ_P0. rewrite -morphV // -!morphM ?(nQt, groupM) ?groupV // ?nQ_P0 //= -Ds2. by rewrite 2!mulgA mulgK mulgKV mulgV morph1. pose x := (y ^ s3)^-1 * y ^ s^-1 * (y ^ (s * s1)^-1)^-1 * y. have{abP0} Dx: x ^ s^-1 = x. rewrite 3!conjMg !conjVg -!conjgM -!invMg (mulgA s) -(expgS _ 1). rewrite [x]abQP0 ?memQP0 // [rhs in y * rhs]abQP0 ?memQP0 //. apply/(canRL (mulKVg _)); rewrite 3!(mulgA y^-1) [RHS]mulgA; congr (_ * _). rewrite [RHS]abQP0 ?memQP0 //; apply/(canRL (mulgK _))/eqP. rewrite -3![eqbLHS]mulgA. rewrite [rhs in y^-1 * rhs]abQP0 ?memQP0 // -eq_invg_sym eq_invg_mul. apply/eqP; transitivity (t ^+ 2 * s1 * (t^-1 * s2 * t^-1) * s3); last first. by rewrite -[s2]invgK -!invMg mulgA Ds2 -(mulgA s3) invMg mulKVg mulVg. rewrite (canRL (mulKg _) Ds312) -2![_ * t^-1]mulgA. have Dt1 si: si \in P0 -> t^-1 = (s^-1 ^ si) ^ y. by move=> P0si; rewrite {2}/conjg -conjVg -(abP0 si) ?groupV ?mulKg. rewrite {1}(Dt1 s1) // (Dt1 s3^-1) ?groupV // -conjXg /conjg. by rewrite !mulgA !invgK !invMg !invgK !mulgA !mulgKV mulg1. have{Dx memQP0} Dx1: x = 1. apply/set1P; rewrite -set1gE; have [_ _ _ <-] := dprodP defQ. rewrite setIAC (setIidPr sQP0_Q) inE -{2}defP0 -cycleV cent_cycle. by rewrite (sameP cent1P commgP) commgEl Dx mulVg eqxx !memQP0. pose t1 := s1 ^ y; pose t3 := s3 ^ y. have{x Dx1} Ds13: s1 * (t * t1)^-1 = (t3 * t)^-1 * s3. by apply/eqP; rewrite eq_sym eq_mulVg1 invMg invgK -Dx1 /x /conjg !gnorm. suffices Ds1: s1 = s^-1. rewrite -(canLR (mulKg _) (canRL (mulgKV _) Dusv1)) Ds1 Duv1. by rewrite !invMg invgK /conjg !gnorm. have [_ _ /trivgPn[u Uu nt_u] _ _] := Frobenius_context frobH. apply: (conjg_inj y); apply: contraNeq nt_u. rewrite -/t1 conjVg -/t eq_mulVg1 -invMg => nt_tt1. have Hu := sUH u Uu; have P1tt1: t * t1 \in P1 by rewrite groupM ?memJ_conjg. have /tiH_P1 defU: (t * t1)^-1 \in P1^# by rewrite 2!inE nt_tt1 groupV. suffices: (u ^ s1) ^ (t * t1)^-1 \in U. rewrite -mem_conjg nUP1 // conjgE mulgA => /(not_splitU _ _ Uu). by rewrite groupV (negPf nt_s1) andbF mulVg eqxx andbT /= => /(_ _ _)/eqP->. rewrite -defU inE memJ_conjg -conjgM Ds13 conjgM groupJ ?(groupJ _ Hs1) //. by rewrite sUH // -mem_conjg nUP1 // groupM ?memJ_conjg. Qed.
|
Fact
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
BGappendixC3_Ediv
| |
BGappendixC_inner_subproof : (p <= q)%N. Proof. have [y QP0y nUP0y] := nU_P0QP0. by apply: Einv_gt1_le_pq E_gt1; apply: BGappendixC3_Ediv nUP0y. Qed.
|
Fact
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
BGappendixC_inner_subproof
| |
prime_dim_normed_finField : (p <= q)%N. Proof. apply: wlog_neg; rewrite -ltnNge => ltqp. have [F sigma /isomP[inj_sigma im_sigma] defP0] := fieldH. case=> sigmaU inj_sigmaU sigmaJ. have oF: #|F| = (p ^ q)%N by rewrite -cardsT -im_sigma card_injm. have charFp: p \in [char F] := card_finCharP oF pr_p. have sP0P: P0 \subset P by rewrite -defP0 subsetIl. pose s := invm inj_sigma 1%R. have sigma_s: sigma s = 1%R by rewrite invmK ?im_sigma ?inE. have{} defP0: <[s]> = P0. by rewrite -morphim_cycle /= ?im_sigma ?inE // morphim_invmE. exact: BGappendixC_inner_subproof defP0 sigmaJ. Qed.
|
Theorem
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.",
"From mathcomp Require Import choice fintype tuple finfun bigop prime finset.",
"From mathcomp Require Import binomial order.",
"From mathcomp Require Import fingroup morphism automorphism quotient action.",
"From mathcomp Require Import gproduct.",
"From mathcomp Require Import ssralg finalg zmodp poly polydiv ssrnum matrix.",
"From mathcomp Require Import mxalgebra vector.",
"From mathcomp Require Import cyclic gfunctor commutator pgroup abelian.",
"From mathcomp Require Import frobenius.",
"From mathcomp Require Import falgebra fieldext galois algC finfield.",
"From mathcomp Require Import mxabelem classfun character integral_char inertia.",
"From odd_order Require Import BGsection1."
] |
theories/BGappendixC.v
|
prime_dim_normed_finField
| |
plength_1 p (G : {set gT}) := 'O_{p^', p, p^'}(G) == G.
|
Definition
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
"From mathcomp Require Import mxalgebra.",
"From mathcomp Require Import cyclic center gfunctor commutator finmodule.",
"From mathcomp Require Import gseries pgroup nilpotent sylow abelian maximal.",
"From mathcomp Require Import hall extremal mxrepresentation mxabelem."
] |
theories/BGsection1.v
|
plength_1
| |
p_elt_gen p (G : {set gT}) := <<[set x in G | p.-elt x]>>.
|
Definition
|
theories
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
"From mathcomp Require Import mxalgebra.",
"From mathcomp Require Import cyclic center gfunctor commutator finmodule.",
"From mathcomp Require Import gseries pgroup nilpotent sylow abelian maximal.",
"From mathcomp Require Import hall extremal mxrepresentation mxabelem."
] |
theories/BGsection1.v
|
p_elt_gen
|
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